Question

Observe the pattern how the bricks are arranged
In the same pattern if 165 bricks were arranged such that 18 bricks are arranged at the base, In how many are all bricks placed? How many bricks are in the top row?

Answer 1

Number of rows = 15

Bricks in top row = 4

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Answer 2

Given:

Total number of bricks are 165 and number of bricks in the base is 18

To find:

Number of rows in which all the bricks placed and number of bricks in the top row

Step-by-step explanation:

The pattern given are forming an A.P.

As given

First term; a= 18

Common difference;d= -1

Sum of all bricks $S_n$ = 165

Now as we know

$S_n=\dfrac{n}{2} (2a+(n-1)d)$

So we have

$165=\dfrac{n}{2} (2\times 18+(n-1)(-1))\\\\\Rightarrow 330 = n(36-n+1)\\\\\Rightarrow 330=n(37-n)\\\\\Rightarrow 330=37n-n^2\\\\\Rightarrow n^2-37n+ 330 =0 \\\\\Rightarrow n^2-15n-22n+330=0 \\\\\Rightarrow n(n-15)-22(n-15)=0\\\\\Rightarrow (n-15)(n-22)=0$

therefore n= 15 or n=22

Now if n is 15

Number of bricks in last row is given by

$a_n=a+(n-1)d\\\\\Rightarrow a_{15}=18+(15-1)(-1)\\\\\Rightarrow a_{15}=18-14=4$

Now if n is 22

Number of bricks in last row

$a_n=a+(n-1)d\\\\\Rightarrow a_{22}=18+(22-1)(-1)\\\\\Rightarrow a_{15}=18-21=-3$

which is not possible

Hence, there are 15 rows and there and 4 bricks in last row